Method of forming a hardened skin on a surface of a molded article

ABSTRACT

A method of forming a hardened skin on one or more surfaces of a molded article. In an exemplary method, a formable material is mixed with a blowing agent to form a foam material. The foam material is placed in a flow molding apparatus such that a first surface of the foam material is in contact with a first mold section and a second surface of the foam material is in contact with a second mold section. In operation, an alternating dielectric field is applied across the foam material to form the molded article. At the end of the molding cycle, the first and/or second surfaces of the foam material remain under the decomposition temperature of the blowing agent and are not blown so as to form one or more thicknesses of hardened skin on the molded article.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of, and claims priority to,U.S. patent application Ser. No. 10/890,906, filed on Jul. 14, 2004, nowU.S. Pat. No. 7,837,910 which is incorporated herein by reference in itsentirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not applicable.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention is generally directed to the field of flowmolding, and is more specifically directed to a unique flow moldingprocess for forming a hardened skin on one or more surfaces of a moldedarticle.

2. Description of Related Art

Various flow molding apparatuses are known in the art that employdielectric heating to mold a plastic part from a formable plasticmaterial. In all of these apparatuses, the plastic material is placedbetween two electrodes such that the material effectively becomes thedielectric of a capacitor. An alternating electric field generatedbetween the electrodes causes polar molecules in the plastic material tobe attracted and repelled by the rapidly changing polarity of theelectric field. The friction resulting from this molecular movementcauses the plastic material to heat throughout its entire mass tothereby form the molded article.

One flow molding apparatus known in the art for making plastic partscomprises a top electrode and a bottom electrode with top and bottommolds disposed therebetween. The top and bottom molds define a moldingcavity in which a plastic material may be placed. Preferably, thecurrent field lines are perpendicular to the plastic material at allpoints along its surface to thereby provide a uniform temperaturethroughout the material. In addition, the top and bottom electrodessubstantially match the configuration of the plastic part that is beingfabricated such that the distance between the electrodes is constant inorder to provide uniform heating of the plastic material. In operation,an alternating electric field is applied across the molding cavity tothereby form the plastic part. An example of this type of a flow moldingapparatus is disclosed in U.S. Pat. No. 4,268,238.

Another flow molding apparatus known in the art for making plastic partscomprises a top electrode and a bottom electrode with a mold disposedtherebetween. The mold has a non-uniform thickness so as to allow themolding of a non-uniform plastic part from a plastic material placedbetween the mold and the top electrode. In order to provide uniformheating throughout the plastic material, a constant capacitance ismaintained throughout all of the different thickness sections of theplastic part. This may be accomplished by equalizing the relativedielectric constants between the plastic material and the mold,preferably by altering the relative dielectric constant of the mold viathe use of additives. Alternatively, the capacitance may be equalized bymodifying the spacing between the top and bottom electrodes in thedifferent thickness sections of the plastic part. An example of thistype of a flow molding apparatus is disclosed in U.S. Pat. No.4,441,876.

Another flow molding apparatus known in the art for making foamedplastic parts comprises a top electrode and a bottom electrode with amold disposed therebetween. A plastic foam material may be placed in acavity of the mold and then compressed during the molding cycle. Afterthe heat is terminated, the compressed plastic foam material ispermitted to expand as it cools so as to conform to the shape of themold and thereby form the foamed plastic part. An example of this typeof a flow molding apparatus is disclosed in U.S. Pat. No. 4,524,037.

Yet another flow molding apparatus known in the art for making foamedplastic parts comprises a top electrode and a bottom electrode with atwo-piece mold disposed therebetween. The mold supports a diaphragm suchthat a plastic foam material may be placed between the diaphragm and thebottom mold. A fluid is injected into the mold above the diaphragm so asto initially deflect the diaphragm and thus expel substantially all ofthe air from the mold. The fluid is then extracted from the mold duringthe molding cycle, which causes a vacuum in the mold to thereby assistin the expansion of the plastic foam material. An example of this typeof an apparatus is disclosed in U.S. Pat. No. 4,851,167.

The flow molding apparatuses and related methods described above aresuitable for the manufacture of many different types of plastic parts,including foamed plastic parts. Many of the foamed plastic parts made inaccordance with these methods, however, are not sufficiently durable towithstand the abrasion that occurs during normal use of the parts, arenot easily washable, cannot be texturized as desired, and/or do notinclude a non-skid surface as required for particular applications.Thus, there is a need in the art for a molding process that overcomesone or more of the problems associated with the methods for forming thefoamed plastic parts described above.

BRIEF SUMMARY OF THE INVENTION

The present invention is directed to a method of forming a hardened skinon one or more surfaces of a molded article. In accordance with thismethod, a formable material is mixed with a blowing agent to form a foammaterial. The foam material is then placed in a flow molding apparatuswhereby an alternating dielectric field is applied across the foammaterial to form the molded article. Typically, most of the foammaterial exceeds the decomposition temperature of the blowing agent andis fully blown at the end of the molding cycle. However, one or moresurfaces of the foam material remain under the decomposition temperatureof the blowing agent and are not blown so as to form one or morethicknesses of hardened skin on the molded article. The method of thepresent invention enables the fabrication of molded articles withhardened skins that are sufficiently durable to withstand the abrasionthat occurs during normal use of the articles, are easily washable, maybe texturized as desired, and/or include a non-skid surface as requiredfor particular applications.

In a first exemplary embodiment, the foam material is placed in a flowmolding apparatus such that the top surface of the foam material is incontact with a top mold of the apparatus and the bottom surface of thefoam material is in contact with a bottom mold of the apparatus. The topmold is modified (e.g., by adjusting its power factor via the use of oneor more suitable additives) so that its bottom surface reaches atemperature that is substantially the same as the molding temperature ofthe foam material at the end of the molding cycle. By matching thetemperature of the top mold to the molding temperature of the foammaterial, the top surface of the foam material exceeds the decompositiontemperature of the blowing agent and is fully blown at the end of themolding cycle. As such, a hardened skin is not formed on the top surfaceof the molded article.

In this embodiment, a bottom electrode of the flow molding apparatus ischilled so as to reduce the temperature of the bottom mold. By doing so,the bottom surface of the foam material remains under the decompositiontemperature of the blowing agent and is not blown at the end of themolding cycle. As such, a hardened skin is formed on the entire bottomsurface of the molded article. The thickness of the hardened skin isdependent on a variety of factors, including the thermal conductivity ofthe bottom mold, the power factor of the bottom mold, and thetemperature of the bottom mold. One or more of these factors may beadjusted in order to increase or decrease the thickness of the hardenedskin as desired for a particular type of molded article.

In a second exemplary embodiment, the foam material is placed in a flowmolding apparatus such that the top surface of the foam material is incontact with a top mold of the apparatus and the bottom surface of thefoam material is in contact with a bottom mold of the apparatus. The topmold is modified such that a hardened skin is not formed on the topsurface of the molded article (as in the first exemplary embodiment).However, in this embodiment, the bottom mold includes different moldsections that are modified to form two different thicknesses of hardenedskin on various bottom surfaces of the molded article. Specifically, afirst thickness of hardened skin is formed on certain bottom surfaces ofthe molded article (i.e., the surfaces in contact with a first group ofmold sections) and a second thickness of hardened skin is formed onother bottom surfaces of the molded article (i.e., the surfaces incontact with a second group of mold sections). As in the first exemplaryembodiment, the thickness of each hardened skin is dependent on avariety of factors, including the thermal conductivity of the bottommold section, the power factor of the bottom mold section, and thetemperature of the bottom mold section. One or more of these factors maybe adjusted in order to increase or decrease the thickness of eachhardened skins as desired for a particular type of molded article.

In general, the method of the present invention may be used to make amolded article from a single foam material placed between the top andbottom molds of a flow molding apparatus. Alternatively, the method maybe used to make a molded article from two or more different formablematerials (at least one of which is a foam material) placed between thetop and bottom molds of a flow molding apparatus. In either case, ahardened skin may be formed on one or more surfaces of the moldedarticle as desired. Accordingly, the method may be used to make avariety of different types of molded articles in accordance with thepresent invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The method of the present invention will be described in greater detailin the following detailed description of the invention with reference tothe accompanying drawings that form a part hereof, in which:

FIG. 1 is a diagram of a flow molding apparatus, wherein a foam materialhas been placed between a top mold and a bottom mold of the apparatusfor making a molded article with a hardened skin on the bottom surfacethereof in accordance with a first exemplary embodiment of the method ofthe present invention;

FIG. 2 is an enlarged view of the foam material in relation to the topand bottom molds of the flow molding apparatus of FIG. 1, wherein athickness x of the foam material remains under the decompositiontemperature of the blowing agent in the foam material to thereby formthe hardened skin on bottom surface of the molded article;

FIG. 3 is a graphical representation showing the relationship betweenthe temperature (T) versus the power factor (pf) for each of the topmold, the foam material, and the bottom mold of the flow moldingapparatus of FIG. 1;

FIG. 4 is a graphical representation showing the relationship betweenthe temperature (T) versus the relative dielectric constant (c) for eachof the top mold, the foam material, and the bottom mold of the flowmolding apparatus of FIG. 1;

FIG. 5 is a graphical representation showing the relationship betweenthe temperature (T) versus the specific heat (h) for each of the topmold, the foam material, and the bottom mold of the flow moldingapparatus of FIG. 1;

FIG. 6 is a diagram of a flow molding apparatus, wherein a foam materialhas been placed between a top mold and a bottom mold of the apparatusfor making a molded article with different thicknesses of hardened skinon the bottom surface thereof in accordance with a second exemplaryembodiment of the method of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention is directed to a method of forming a hardened skinon one or more surfaces of a molded article. In accordance with thismethod, a formable material is mixed with a blowing agent to form a foammaterial. The foam material is then placed between a top mold and abottom mold of a flow molding apparatus whereby an alternatingdielectric field is applied across the foam material to form the moldedarticle. The temperature of certain mold sections are reduced so thatone or more surfaces of the foam material will remain under thedecomposition temperature of the blowing agent to thereby form ahardened skin on the molded article. One skilled in the art willunderstand that the method of the present invention may be used tomanufacture a variety of different types of molded articles for use in avariety of different industries.

As used herein, the term “foam material” means any formable materialmixed with one or more blowing agents that may be heated to form adefined shape within a mold of a flow molding apparatus. As is known inthe art, many different types of formable materials (includingthermoplastics and thermosets) and many different types of blowingagents may be used to form a foam material and will vary depending onthe molded article to be fabricated. It may also be desirable to mix oneor more cross-linkers with the foam material in order to provide greatermaterial strength. Examples of suitable foam materials includecross-linked PE-EVA foam, PVC foam, vinyl nitrile foam, neoprene foam,foamed rubber, polypropylene, and blends of any of the foregoing. Ofcourse, it should be understood that many other types of foam materialsmay also be used in accordance with the present invention.

It should be understood that each of the foregoing foam materials has amolding temperature and molding time associated therewith. As usedherein, the term “molding temperature” means the temperature at which afoam material is blown and/or cross-linked (which may comprise anytemperature above the decomposition temperature of the blowing agent inthe foam material), and the term “molding time” means the amount of timerequired for a foam material to reach its molding temperature. It willbe seen that any portion of a foam material that does not reach itsmolding temperature and remains under the decomposition temperature ofthe blowing agent will not be blown and, thus, will form a hardened skinon the surface of the molded article.

FIRST EXEMPLARY EMBODIMENT

Referring to FIG. 1, a first exemplary embodiment of the method of thepresent invention will now be described with reference to the diagram ofa flow molding apparatus designated generally as numeral 10. Flowmolding apparatus 10 includes a top electrode 12 and a bottom electrode14, both of which are connected to an electromagnetic energy source (notshown) operable to generate an alternating electric field between theelectrodes. The alternating electric field may be generated atfrequencies ranging from 1 MHz to 500 MHz, is preferably generated atfrequencies ranging from 10 MHz to 100 MHz, and is most preferablygenerated at either 26 MHz or 40 MHz. Also included within apparatus 10are a top mold 16 and a bottom mold 18 that together define a moldingcavity therebetween.

In the illustrated example, a foam material 20 is placed within themolding cavity such that a top surface of foam material 20 is in contactwith top mold 16 and a bottom surface of foam material 20 is in contactwith bottom mold 18. Of course, it should be understood that additionalformable materials could also be placed within the molding cavity asdescribed in Applicant's co-pending patent application entitled “Methodof Making a Molded Article from Two or more Different Formable Materialsin a Single Heating Cycle” (U.S. patent application Ser. No.10/890,904), which is incorporated herein by reference. In operation, analternating dielectric field is applied across foam material 20 tothereby form the molded article.

It should be understood that flow molding apparatus 10 is merely anexample of an apparatus that may be used to make a molded article inaccordance with the method of the present invention. Other flow moldingapparatuses and related methods may also be used, such as thosedisclosed in U.S. Pat. No. 4,268,238, U.S. Pat. No. 4,441,876, U.S. Pat.No. 4,524,037 and U.S. Pat. No. 4,851,167, all of which are incorporatedherein by reference.

Referring still to FIG. 1, it can be seen that there are three layers ofmaterial between top electrode 12 and bottom electrode 14, namely, topmold 16 (layer 1), foam material 20 (layer 2), and bottom mold 18 (layer3). The following general equations may be established with respect tothese three layers of material (which will be used hereinbelow todescribe the method of the present invention). It should be understoodthat the subscript 1 in these equations denotes the layer number of theparticular material (i.e., subscript 1 denotes top mold 16 (layer 1),subscript 2 denotes foam material 20 (layer 2), and subscript 3 denotesbottom mold 18 (layer 3)).

First, the capacitance of each layer of material may be expressed by thefollowing equation:

$\begin{matrix}{C_{i} = \frac{25.4 \times ɛ_{i} \times A_{i}}{36 \times \pi \times d_{i}}} & (1)\end{matrix}$where

-   -   C_(i)=capacitance of layer i in picofarads    -   ∈_(i)=relative dielectric constant of layer i    -   A_(i)=area of layer i in inches²    -   d_(i)=thickness of layer i in inches.

The equivalent capacitance of all three layers of material is given bythe following equation:

$\begin{matrix}{{Ceq} = \frac{C_{1} \times C_{2} \times C_{3}}{\left( {C_{1} \times C_{2}} \right) + \left( {C_{1} \times C_{3}} \right) + \left( {C_{2} \times C_{3}} \right)}} & (2)\end{matrix}$where

-   -   C_(eq)=equivalent capacitance of layers in picofarads    -   C₁=capacitance of layer 1 in picofarads    -   C₂=capacitance of layer 2 in picofarads    -   C₃=capacitance of layer 3 in picofarads.

The equivalent reactance associated with the equivalent capacitance ofall three layers of material may then be given by the followingequation:

$\begin{matrix}{X_{eq} = \frac{1}{2 \times \pi \times f \times C_{eq}}} & (3)\end{matrix}$where

-   -   X_(eq)=equivalent reactance of layers in ohms    -   f=frequency of dielectric field in hertz    -   C_(eq)=equivalent capacitance of layers in farads.

The resistance of each layer of material is equal to the product of thepower factor of that layer and the equivalent reactance. Therefore,using the equivalent reactance from equation (3), the resistance of eachlayer of material may be expressed as follows:

$\begin{matrix}{R_{i} = \frac{{pf}_{i}}{2 \times \pi \times f \times C_{i}}} & (4)\end{matrix}$where

-   -   R_(i)=resistance of layer i in ohms    -   pf_(i)=power factor of layer i    -   f=frequency of dielectric field in hertz    -   C_(i)=capacitance of layer i in farads.

Next, the current passing between top electrode 12 and bottom electrode14 through all three layers of material may be represented by thefollowing equation:

$\begin{matrix}{I = \frac{V}{\sqrt{X_{eq}^{2} + R_{eq}^{2}}}} & (5)\end{matrix}$where

-   -   I=current in amperes    -   V=voltage between the electrodes in volts    -   X_(eq)=equivalent reactance of layers in ohms    -   R_(eq)=equivalent resistance of layers in ohms.

Assuming that the equivalent resistance of all three layers of materialis small compared to the equivalent reactance, equation (5) may besimplified as follows:

$\begin{matrix}{I = \frac{V}{X_{eq}}} & (6)\end{matrix}$

Furthermore, the power that is dissipated in each layer of material dueto the application of the dielectric field may be expressed by thefollowing equation:P _(i) =R _(i) ×I ²  (7)where

-   -   P_(i)=power of layer i in watts due to the dielectric field    -   R_(i)=resistance of layer i in ohms    -   I=current in amperes.

By combining equations (4) and (7), the power that is dissipated in eachlayer of material due to the application of the dielectric field may beexpressed as follows:

$\begin{matrix}{P_{i} = \frac{{pf}_{i} \times I^{2}}{2 \times \pi \times f \times C_{i}}} & (8)\end{matrix}$

Now, the increase in temperature of each layer of material during themolding cycle may be represented by the following equation:

$\begin{matrix}{{\Delta\; T_{i}} = \frac{P_{i} \times t_{i}}{16.387 \times h_{i} \times \rho_{i} \times d_{i}}} & (9)\end{matrix}$where

-   -   ΔT_(i)=increase in temperature of layer i in degrees Celsius    -   P_(i)=power of layer i in watts due to the dielectric field    -   t_(i)=molding time of layer i in seconds    -   h_(i)=specific heat of layer i    -   ρ_(i)=specific gravity of layer i    -   d_(i)=thickness of layer i in inches.

By combining equations (8) and (9), the increase in temperature of eachlayer of material during the molding cycle may be expressed as follows:

$\begin{matrix}{{\Delta\; T_{i}} = \frac{{pf}_{i} \times I^{2} \times t_{i}}{2 \times \pi \times f \times C_{i} \times 16.387 \times h_{i} \times \rho_{i} \times d_{i}}} & (10)\end{matrix}$

By solving equation (10) for t_(i), the molding time for each layer ofmaterial may be expressed by the following equation:

$\begin{matrix}{t_{i} = \frac{16.387 \times \Delta\; T_{i} \times h_{i} \times \rho_{i} \times d_{i}}{\frac{{pf}_{i} \times I^{2}}{2 \times \pi \times f \times C_{i}}}} & (11)\end{matrix}$

Equation (11) may then be solved for pf_(i) such that the power factorfor each layer of material may be expressed as follows:

$\begin{matrix}{{pf}_{i} = \frac{16.387 \times \Delta\; T_{i} \times h_{i} \times \rho_{i} \times d_{i} \times 2 \times \pi \times f \times C_{i}}{t_{i} \times I^{2}}} & (12)\end{matrix}$

In accordance with the method of the present invention, a hardened skinmay be formed on one or more surfaces of foam material 20. In this firstexemplary embodiment, a hardened skin is formed on the entire bottomsurface of foam material 20 (although it should be understood that ahardened skin could also be formed on the entire top surface of foammaterial 20 if desired). Accordingly, the following steps may beperformed using the method of the present invention: (1) calculate themolding time of foam material 20; (2) modify top mold 16 so that the topsurface of foam material 20 will be fully blown (and preferablycross-linked) at the end of the molding cycle; and (3) reduce thetemperature of bottom mold 18 so that a hardened skin is formed on theentire bottom surface of foam material 20 at the end of the moldingcycle. Each of these steps will be described in detail hereinbelow.

Step 1: Calculate Molding Time of Foam Material

First, equation (11) may be used to calculate the molding time (t₂) offoam material 20 (i.e., the time that it takes for foam material 20 toreach its molding temperature). It can be seen from equation (11) thatthe molding time (t₂) of foam material 20 is a function of the followingfactors: the increase in temperature (ΔT₂) required for foam material 20to reach its molding temperature; the specific heat (h₂) of foammaterial 20; the specific gravity (ρ₂) of foam material 20; thethickness (d₂) and capacitance (C₂) of foam material 20; the powerfactor (pf₂) of foam material 20; the current (I) passing between topelectrode 12 and bottom electrode 14 (which may be calculated fromequation (6)); and the frequency (f) of the dielectric field.

Step 2: Modify Top Mold so that Top Surface of Foam Material Will beFully Blown

Second, the molding time (t₂) of foam material 20 (as calculated instep 1) is used to calculate a required power factor (pf₁) for top mold16. As stated above, the temperature of top mold 16 is desirably chosenso that the top surface of foam material 20 will be fully blown at theend of the molding cycle. As such, the required power factor (pf₁) fortop mold 16 will be the power factor that allows the bottom surface oftop mold 16 to reach a temperature that is substantially the same as themolding temperature of foam material 20 at the end of the molding cycle.

Equation (12) may be used to calculate the required power factor (pf₁)for top mold 16. It can be seen from equation (12) that the requiredpower factor (pf₁) for top mold 16 is a function of the followingfactors: the increase in temperature (ΔT₁) required for top mold 16 toreach the molding temperature of foam material 20; the specific heat(h₁) of top mold 16; the specific gravity (ρ₁) of top mold 16; thethickness (d₁) and capacitance (C₁) of top mold 16; the frequency (f) ofthe dielectric field; the molding time (t₂) of foam material 20; and thecurrent (I) passing between top electrode 12 and bottom electrode 14(which may be calculated from equation (6)).

Next, the power factor of top mold 16 is adjusted to match its requiredpower factor. The power factor of top mold 16 may be adjusted by variousmeans known in the art, and is preferably adjusted by selecting anadditive, calculating an amount of the additive to be mixed with thematerial of top mold 16 so that the power factor substantially matchesthe required power factor, and then mixing the calculated amount ofadditive with the material of top mold 16. It should be understood thata selected additive may be used to increase or decrease the power factorof the material and, preferably, does not otherwise alter the propertiesof the material. It may also be desirable to use a mixture of two ormore additives depending on the power factor of each of the additives asa function of temperature.

Once an additive(s) is selected for top mold 16, the following generalequations may be used to calculate the amount of the additive to bemixed with the material of top mold 16 so that the power factorsubstantially matches the required power factor. First, the power factorequivalent for the material/additive mixture may be expressed by thefollowing equation:

$\begin{matrix}{{pf}_{eq} = \frac{\begin{matrix}{\left( {{pf}_{material} \times d_{material} \times ɛ_{additive}} \right) +} \\\left( {{pf}_{additive} \times d_{additive} \times ɛ_{material}} \right)\end{matrix}}{\left( {ɛ_{material} \times d_{additive}} \right) + \left( {ɛ_{additive} + d_{material}} \right)}} & (13)\end{matrix}$where

-   -   pf_(eq)=power factor equivalent of the material/additive mixture    -   pf_(material)=power factor of the material to be modified    -   pf_(additive)=power factor of the selected additive    -   d_(material)=thickness of the material to be modified    -   d_(additive)=thickness of the selected additive    -   ∈_(material)=relative dielectric constant of the material to be        modified    -   ∈_(additive)=relative dielectric constant of the selected        additive.

Now, assume x is the percentage of the mixture by volume comprising theadditive and (100−x) is the percentage of the mixture by volumecomprising the material. Substituting x for d_(additive) and (100−x) ford_(material), equation (13) may be rewritten as follows:

$\begin{matrix}{{pf}_{eq} = \frac{\begin{matrix}{\left( {{pf}_{material} \times \left( {100 - x} \right) \times ɛ_{additive}} \right) +} \\\left( {{pf}_{additive} \times x \times ɛ_{material}} \right)\end{matrix}}{\left( {ɛ_{material} \times x} \right) + \left( {ɛ_{additive} \times \left( {100 - x} \right)} \right)}} & (14)\end{matrix}$

Equation (14) may then be solved for x and rewritten as follows:

$\begin{matrix}{x = \frac{\left( {100 \times {pf}_{material} \times ɛ_{additive}} \right) - \left( {100 \times {pf}_{eq} \times ɛ_{additive}} \right)}{\begin{matrix}{\left( {{pf}_{eq} \times ɛ_{{material}\;}} \right) - \left( {{pf}_{eq} \times ɛ_{additive}} \right) +} \\{\left( {{pf}_{material} \times ɛ_{additive}} \right) - \left( {{pf}_{additive} \times ɛ_{material}} \right)}\end{matrix}}} & (15)\end{matrix}$

Thus, equation (15) may be used to calculate the amount of the selectedadditive to be mixed with the material of top mold 16 so that the powerfactor equivalent (pf_(eq)) of the mixture substantially matches therequired power factor (pf₁) of top mold 16 calculated above (i.e.,pf_(eq)=pf₁). As such, the bottom surface of top mold 16 will reach atemperature that is substantially the same as the molding temperature offoam material 20 whereby the top surface of foam material 20 will befully blown at the end of the molding cycle. Accordingly, a hardenedskin will not be formed on the top surface of the molded article.

Step 3: Reduce Temperature of Bottom Mold to Form Hardened Skin onBottom Surface of Foam Material

Third, the molding time (t₂) of foam material 20 (as calculated instep 1) is used to calculate the increase in temperature (ΔT₃) of bottommold 18 at the end of the molding cycle. Preferably, bottom electrode 14is chilled throughout the molding cycle so that the temperature ofbottom mold 18 may be substantially reduced to enable the formation of ahardened skin on the bottom surface of the molded article. Equation (10)may be used to calculate the increase in temperature (ΔT₃) of bottommold 18, which is a function of the following factors: the power factor(pf₃) of bottom mold 18; the current (I) passing between top electrode12 and bottom electrode 14 (which may be calculated from equation (6));the molding time (t₂) of foam material 20; the frequency (f) of thedielectric field; the specific heat (h₃) of bottom mold 18; the specificgravity (ρ₃) of bottom mold 18; and the thickness (d₃) and capacitance(C₃) of bottom mold 18.

Because the temperature of bottom mold 18 will be less than thetemperature of foam material 20, a certain amount of heat will betransferred from foam material 20 to bottom mold 18 during the moldingcycle. The amount of heat exchange will depend on the thermalconductivity (k₂) of foam material 20 and the thermal conductivity (k₃)of bottom mold 18. Specifically, the heat gained or lost by foammaterial 20 is given by the following equation:Q ₂ =k ₂(T _(2,3) −T ₂)  (16)where

-   -   Q₂=power gained/lost by layer 2 in watts due to heat exchange        with layer 3    -   k₂=thermal conductivity of layer 2 in watts/inches²/inch/° C.    -   T₂=temperature of layer 2 in degrees Celsius    -   T_(2,3)=temperature at the interface of layers 2 and 3 in        degrees Celsius

Similarly, the heat gained or lost by bottom mold 18 is given by thefollowing equation:Q ₃ =k ₃(T _(2,3) −T ₃)  (17)where

-   -   Q₃=power gained/lost by layer 3 in watts due to heat exchange        with layer 2    -   k₃=thermal conductivity of layer 3 in watts/inches²/inch/° C.    -   T_(2,3)=temperature at the interface of layers 2 and 3 in        degrees Celsius    -   T₃=temperature of layer 3 in degrees Celsius

Because the heat gained or lost by foam material 20 (Q₂) is equal to theheat gained or lost by bottom mold 18 (Q₃) such that Q₂+Q₃=0, equations(16) and (17) may be combined as follows:k ₂(T _(2,3) −T ₂)+k ₃(T _(2,3) −T ₃)=0  (18)

Equation (18) may then be solved for T_(2,3) such that the temperatureat the interface between foam material 20 and bottom mold 18 is given bythe following equation:

$\begin{matrix}{T_{2,3} = \frac{{k_{2}T_{2}} + {k_{3}T_{3}}}{k_{2} + k_{3}}} & (19)\end{matrix}$

Now, referring to FIG. 2, an enlarged view of foam material 20 inrelation to top mold 16 and bottom mold 18 is provided wherein acenterline 22 has been drawn through foam material 20. Because the totalthickness of foam material 20 is d₂, the thickness of foam materialbetween centerline 22 and the interface of foam material 20 with bottommold 18 is d₂/2. Of course, the thickness of foam material betweencenterline 22 and the interface of foam material 20 with top mold 16would also be d₂/2.

At the end of the molding cycle, the temperature of foam material 20 atcenterline 22 will be the molding temperature (T₂) of foam material 20.However, the temperature of foam material 20 below centerline 22 willvary from the molding temperature (T₂) of foam material 20 (i.e., thetemperature at centerline 22) to the temperature (T_(2,3)) derived fromequation (19) above (i.e., the temperature at the interface between foammaterial 20 and bottom mold 18). A temperature line 24 drawn fromcenterline 22 to the interface of foam material 20 with bottom mold 18will have a slope represented by the following equation:

$\begin{matrix}{{slope} = \frac{T_{2} - T_{2,3}}{\frac{d_{2}}{2}}} & (20)\end{matrix}$where

-   -   slope=slope of temperature line    -   T₃=temperature of layer 3 in degrees Celsius    -   T_(2,3)=temperature at the interface of layers 2 and 3 in        degrees Celsius    -   d₂=thickness of layer 2 in inches

It should be understood that the blowing agent in foam material 20 willhave a decomposition temperature (T_(B)) associated therewith, which isless than the molding temperature (T₂) of foam material 20. Assumingthat the decomposition temperature (T_(B)) of the blowing agent isgreater than the temperature (T_(2,3)) at the interface between foammaterial 20 and bottom mold 18 (as shown in FIG. 2), the thickness (x)of foam material 20 that remains under the decomposition temperature(T_(B)) of the blowing agent may be represented by the followingequation:

$\begin{matrix}{x = \frac{T_{B} - T_{2,3}}{slope}} & (21)\end{matrix}$where

-   -   x=thickness of layer 2 that remains under T_(B) in inches    -   T_(B)=decomposition temperature of blowing agent in degrees        Celsius    -   T_(2,3)=temperature at the interface of layers 2 and 3 in        degrees Celsius    -   slope=slope of temperature line

The thickness (x) of foam material 20 that remains under thedecomposition temperature (T_(B)) of the blowing agent will not be blownat the end of the molding cycle and, thus, will form a hardened skin onthe bottom surface of the molded article.

Based on the equations above, it should be understood that the thicknessof the hardened skin formed on the bottom surface of the molded articleis dependent on a variety of factors, including the thermal conductivity(k₃) of bottom mold 18, the power factor (pf₃) of bottom mold 18, andthe temperature of bottom electrode 14. For example, the thickness ofthe hardened skin may be increased by increasing the thermalconductivity (k₃) of bottom mold 18, decreasing the power factor (pf₃)of bottom mold 18, and/or decreasing the temperature of bottom electrode14. Conversely, the thickness of the hardened skin may be decreased bydecreasing the thermal conductivity (k₃) of bottom mold 18, increasingthe power factor (pf₃) of bottom mold 18, and/or increasing thetemperature of bottom electrode 14. Thus, one or more of these factorsmay be adjusted in order to increase or decrease the thickness of thehardened skin as desired for a particular type of molded article.

It should be understood that the analysis set forth above will provide aclose approximation of the values for the molding time (t₂) of foammaterial 20, the required power factor (pf₁) of top mold 16, thetemperature (T₃) of bottom mold 18, and the thickness (x) of foammaterial 20 that remains under the decomposition temperature (T_(B)) ofthe blowing agent (which forms the hardened skin on the bottom surfaceof the molded article). However, if more exact values are desired, it isnecessary to take into account the heat exchange between adjacent layersof material in time over the course of the molding cycle.

The heat generated in a layer of material i due to the adjacent layeri−1 may be expressed as follows:

$\begin{matrix}{Q_{i,{i - 1}} = {\frac{k_{i - 1} \times k_{i}}{k_{i - 1} + k_{i}}\left( {T_{i - 1} - T_{i}} \right)\frac{2}{d_{i}}}} & (22)\end{matrix}$where

-   -   Q_(i, i−1)=power of layer i in watts due to heat exchange from        layer i−1    -   k_(i−1)=thermal conductivity of layer i−1 in        watts/inches²/inch/° C.    -   k_(i)=thermal conductivity of layer i in watts/inches²/inch/° C.    -   T_(i−1)=temperature of layer i−1 in degrees Celsius    -   T_(i)=temperature of layer i in degrees Celsius    -   d_(i)=thickness of layer i in inches

Similarly, the heat generated in a layer of material i due to theadjacent layer i+1 may be expressed as follows:

$\begin{matrix}{Q_{i,{i + 1}} = {\frac{k_{i + 1} \times k_{i}}{k_{i + 1} + k_{i}}\left( {T_{i + 1} - T_{i}} \right)\frac{2}{d_{i}}}} & (23)\end{matrix}$where

-   -   Q_(i, i+1)=power of layer i in watts due to heat exchange from        layer i+1    -   k_(i+1)=thermal conductivity of layer i+1 in        watts/inches²/inch/° C.    -   k_(i)=thermal conductivity of layer i in watts/inches²/inch/° C.    -   T_(i+1)=temperature of layer i+1 in degrees Celsius    -   T_(i)=temperature of layer i in degrees Celsius    -   d_(i)=thickness of layer i in inches

Accordingly, equations (22) and (23) may be combined such that the heatgenerated in a layer of material i due to both of the adjacent layersi−1 and i+1 may be expressed as follows:

$\begin{matrix}{Q_{i} = {\frac{2}{d_{i}}\left\lbrack {{\frac{k_{i - 1} \times k_{i}}{k_{i - 1} + k_{i}}\left( {T_{i - 1} - T_{i}} \right)} + {\frac{k_{i + 1} \times k_{i}}{k_{i + 1} + k_{i}}\left( {T_{i + 1} - T_{i}} \right)}} \right\rbrack}} & (24)\end{matrix}$

Now, by combining equations (8) and (24), the total power dissipated ina layer of material i due to the application of the dielectric field andthe heat exchange between adjacent layers of material may be expressedas follows:

$\begin{matrix}{{P_{i} + Q_{i}} = {\frac{{pf}_{i} \times I^{2}}{2 \times \pi \times f \times C_{i}} + {\frac{2}{d_{i}}\left\lbrack {{\frac{k_{i - 1} \times k_{i}}{k_{i - 1} + k_{i}}\left( {T_{i - 1} - T_{i}} \right)} + {\frac{k_{i + 1} \times k_{i}}{k_{i + 1} + k_{i}}\left( {T_{i + 1} - T_{i}} \right)}} \right\rbrack}}} & (25)\end{matrix}$

One skilled in the art will understand that equation (25) may be used(in place of equation (8)) in connection with the analysis set forthabove to calculate more exact values for the molding time (t₂) of foammaterial 20, the required power factor (pf₁) of top mold 16, thetemperature (T₃) of bottom mold 18, and the thickness (x) of foammaterial 20 that remains under the decomposition temperature (T_(B)) ofthe blowing agent. As will be described in greater detail with referenceto Example 3 below, these calculations are preferably performed atregular time intervals (such as 1 second time intervals) in order todetermine P_(i)+Q_(i) for each of the layers of material at each of thetime intervals. Of course, a computer may be programmed to perform thesecalculations in order to simplify the analysis.

It should also be understood that the analysis set forth above does nottake into account the fact that the power factor, relative dielectricconstant, specific heat and thermal conductivity of each of the layersof material vary with temperature. Therefore, when performing thecalculations at regular time intervals as discussed above, it ispreferable to use the values for the power factor, relative dielectricconstant, specific heat and thermal conductivity that correspond to thetemperature of each of the layers of material at that particular pointin time. By doing so, it is possible to obtain even more exact valuesfor the molding time (t₂) of foam material 20, the required power factor(pf₁) of top mold 16, the temperature (T₃) of bottom mold 18, and thethickness (x) of foam material 20 that remains under the decompositiontemperature (T_(B)) of the blowing agent (which forms the hardened skinon the bottom surface of the molded article).

It should further be understood that the analysis set forth above doesnot consider the exothermic or endothermic reaction that occurs when ablowing agent reaches its decomposition temperature (which, for manyblowing agents, is 150° C.). The change in heat caused by thisexothermic or endothermic reaction varies in time and may be added tothe above equations if desired. Of course, one skilled in the art willappreciate that the impact of the exothermic or endothermic reaction isnot highly significant because it occurs near the end of the moldingcycle and does not involve a large amount of energy.

Three examples will now be provided with reference to the flow moldingapparatus of FIGS. 1 and 2 to further describe the method of the presentinvention. It should be understood that these examples are providedmerely to illustrate the manner in which the method may be used tomanufacture a molded article with a hardened skin and do not in any waylimit the scope of the present invention.

Example 1

Assume for purposes of this example that top mold 16 and bottom mold 18of flow molding apparatus 10 are each formed from silicone rubber V-1008(manufactured by Rhodia Inc.) and foam material 20 comprises an EVA foamblend. Table 1 is provided below to show various factors for each ofthese layers of material, namely, the desired temperature (T) at the endof the molding cycle, thickness (d), power factor (pf), relativedielectric constant (∈), specific heat (h), and specific gravity (ρ).These factors will be used hereinbelow to perform various calculationsin accordance with the method of the present invention.

TABLE 1 Foam Top Mold Material Bottom Mold (layer 1) (layer 2) (layer 3)Desired 200 200 Temperature (T) (degrees Celsius) Thickness (d) .125.375 .125 (inches) Power Factor (pf) .0137 .0033 Relative 3.07 2.67 2.67Dielectric Constant (ε) Specific Heat (h) 1.233 1.566 1.233 SpecificGravity 1.16 1.041 1.16 (ρ)

As can be seen from Table 1, the desired temperature of foam material 20at the end of the molding cycle is its molding temperature of 200° C.(the temperature at which foam material 20 is fully blown andcross-linked). In this example, it is desired that the top surface ofthe molded article be fully blown at the end of the molding cycle. Assuch, the desired temperature of the bottom surface of top mold 16 atthe end of the molding cycle is 200° C. (the molding temperature of foammaterial 20). On the other hand, a hardened skin is desired on thebottom surface of the molded article at the end of the molding cycle. Assuch, bottom electrode 14 is chilled and maintained at 0° C. throughoutthe molding cycle so that the temperature of bottom mold 18 issubstantially reduced. The exact temperature of bottom mold 18 at theend of the molding cycle will be calculated hereinbelow.

Table 1 also shows the various values for the thickness (d) of each ofthe layers of material which, as illustrated in FIG. 1, are shown asbeing constant along the length of the molded article. It should beunderstood, however, that the thickness of foam material 20 may vary atdifferent points along the length of the molded article. In that case,bottom mold 18 would preferably be configured so that its thicknessvaries in such a manner that the sum of the thickness values for foammaterial 20 and bottom mold 18 remain constant. As such, the totaldistance between top electrode 12 and bottom electrode 14 would remainconstant to thereby provide uniform heating of foam material 20 (asdescribed in U.S. Pat. No. 4,268,238). It should also be noted that therelative dielectric constants (∈) of foam material 20 and bottom mold 18would preferably be equalized (as shown in Table 1) so that foammaterial 20 would be uniformly heated (as discussed in U.S. Pat. No.4,441,876).

In addition, Table 1 shows the various values for the power factor (pf),relative dielectric constant (∈), specific heat (h) and specific gravity(ρ) for each of the layers of material. As discussed above, the valuesfor the power factor (pf), relative dielectric constant (∈) and specificheat (h) vary with temperature. For example, FIG. 3 illustrates thetemperature dependence of the power factor (pf) for each of the layersof material, FIG. 4 illustrates the temperature dependence of therelative dielectric constant (∈) for each of the layers of material, andFIG. 5 illustrates the temperature dependence of the specific heat (h)for each of the layers of material. It should be understood that thecurves of FIGS. 3-5 were each integrated over the entire temperaturerange and the resultant averages are shown in Table 1 (which will beused to provide a close approximation of various values in accordancewith the analysis set forth above).

In accordance with the method of the present invention, equations(1)-(21) may be used to calculate the molding time of foam material 20,the required power factor of top mold 16, the temperature of bottom mold18, and the thickness of foam material 20 that remains under thedecomposition temperature of the blowing agent. It should be understoodthat the subscripts used in the following equations denote the layernumber of the particular material (i.e., subscript 1 denotes top mold 16(layer 1), subscript 2 denotes foam material 20 (layer 2), and subscript3 denotes bottom mold 18 (layer 3)).

The capacitance of each of the layers of material may be calculated fromequation (1) using the values for the relative dielectric constant(∈_(i)) and thickness (d₁) shown in Table 1 (assuming that the area ofeach of the layers of material is 1 inch²):

$C_{1} = {\frac{25.4 \times 3.07 \times 1}{36 \times \pi \times {.123}} = {5.429\mspace{11mu}{pF}}}$$C_{2} = {\frac{25.4 \times 2.67 \times 1}{36 \times \pi \times {.375}} = {1.599\mspace{11mu}{pF}}}$$C_{3} = {\frac{25.4 \times 2.67 \times 1}{36 \times \pi \times {.125}} = {4.72\mspace{11mu}{pF}}}$

Next, the equivalent capacitance may be calculated from equation (2)using the capacitance values for each of the layers of material derivedabove:

$C_{eq} = {\frac{5.429 \times 1.599 \times 4.72}{\left( {5.429 \times 1.599} \right) + \left( {5.429 \times 4.72} \right) + \left( {1.599 \times 4.72} \right)} = {0.954\mspace{11mu}{pF}}}$

The equivalent reactance may then be calculated from equation (3) usingthe equivalent capacitance (C_(eq)) derived above (assuming that thefrequency of the electric field is 40 MHz):

$X_{eq} = {\frac{1}{2 \times \pi \times 40 \times 10^{6} \times {.954} \times 10^{- 12}} = {4,171\mspace{14mu}{ohms}}}$

The current passing between top electrode 112 and bottom electrode 114through all of the layers of material may then be calculated fromequation (6) using the equivalent reactance (X_(eq)) derived above(assuming that the voltage is 4,000 volts):

$I = {\frac{4,000}{4,171} = {{.959}\mspace{14mu}{amperes}}}$

First, in accordance with step 1 above, the molding time for foammaterial 20 may be calculated from equation (11) using the current (I)and capacitance (C₂) derived above and the values for the temperature(T₂), thickness (d₂), power factor (pf₂), specific heat (h₂), andspecific gravity (ρ₂) shown in Table 1 (assuming that the startingtemperature is 0° C.):

$t_{2} = {\frac{16.387 \times 200 \times 1.566 \times 1.041 \times {.375}}{\frac{{.0137} \times {.959}^{2}}{2 \times \pi \times 40 \times 10^{6} \times 1.599 \times 10^{- 12}}} = {63.54\mspace{14mu}\sec}}$

Thus, the molding time for foam material 20 is 63.54 seconds (i.e., thetime that it takes foam material 20 to reach its molding temperature of200° C.).

Second, in accordance with step 2 above, a required power factor for topmold 16 may be calculated from equation (12) using the molding time (t₂)for foam material 20, the current (I) and capacitance (C₁) derivedabove, and the values for the temperature (T₁), thickness (d₁), specificheat (h₁), and specific gravity (ρ₁) shown in Table 1 (again, assumingthat the starting temperature is 0° C.):

${pf}_{1} = {\frac{\begin{matrix}{16.387 \times 200 \times 1.233 \times 1.16 \times} \\{{.125} \times 2 \times \pi \times 40 \times 10^{6} \times 5.429 \times 10^{12}}\end{matrix}}{63.54 \times {.959}^{2}} = 0.01368}$

Thus, the required power factor for top mold 16 is 0.01368 (i.e., thepower factor that would allow the bottom surface of top mold 16 to reacha temperature of 200° C. at substantially the same time that foammaterial 20 reaches its molding temperature of 200° C.). It should beunderstood that the power factor of top mold 16 may be adjusted byselecting a polar additive, calculating the amount of the polar additiveto be mixed with the material of top mold 16 so that the power factormatches the required power factor derived above, and then mixing thecalculated amount of the polar additive with the material of top mold16.

Third, in accordance with step 3 above, the temperature of bottom mold18 at the end of the molding cycle may be calculated from equation (10)using the current (I), the calculated molding time (t₂) for foammaterial 20, and the capacitance (C₃) derived above, and the values forthe power factor (pf₃), specific heat (h₃), specific gravity (ρ₃), andthickness (d₃) shown in Table 1:

$T_{3} = {\frac{{.0033} \times 63.54 \times {.959}^{2}}{\begin{matrix}{1.233 \times 1.16 \times {.125} \times 16.387 \times} \\{2 \times \pi \times 40 \times 10^{6} \times 4.72 \times 10^{12}}\end{matrix}} = {55.48{^\circ}\mspace{14mu}{C.}}}$

Thus, the temperature of bottom mold 18 at the end of the molding cycleis 55.48° C. (assuming bottom electrode 14 is chilled and maintained at0° C. throughout the molding cycle).

Next, the temperature at the interface between foam material 20 andbottom mold 18 may be calculated from equation (19) using thetemperature (T₃) of bottom mold 18 derived above and the value for thetemperature (T₂) of foam material 20 shown in Table 1 (assuming that thethermal conductivity of foam material 20 is 0.00638 and the thermalconductivity of bottom mold 18 is 0.00585):

$T_{2,3} = {\frac{\left( {{.00638} \times 200} \right) + \left( {{.00585} \times 55.48} \right)}{{.00638} + {.00585}} = {130.87{^\circ}\mspace{14mu}{C.}}}$

Thus, the temperature at the interface between foam material 20 andbottom mold 18 is 130.87° C.

Equation (20) may then be used to determine the slope of temperatureline 24 using the temperature (T_(2,3)) at the interface between foammaterial 20 and bottom mold 18 derived above and the values for thetemperature (T₂) of foam material 20 and the thickness (d₂) of foammaterial 20 shown in Table 1:

${slope} = {\frac{200 - 130.87}{.1875} = {368.68{^\circ}\mspace{14mu}{{C.}/{inch}}}}$

Thus, the slope of temperature line 24 is 368.68° C./inch.

Now, the thickness (x) of foam material 20 that remains under thedecomposition temperature (T_(B)) of the blowing agent may be calculatedfrom equation (21) using the temperature (T_(2,3)) at the interfacebetween foam material 20 and bottom mold 18 derived above and the slopeof temperature line 24 derived above (assuming that the decompositiontemperature (T_(B)) of the blowing agent in foam material 20 is 150°C.):

$x = {\frac{150 - 130.87}{368.68} = {0.0519\mspace{14mu}{inches}}}$

Thus, the thickness (x) of foam material 20 that remains under thedecomposition temperature (T_(B)) of the blowing agent is 0.0519 inches.As such, a layer of foam material 20 having a thickness of 0.0519 incheswill not be blown at the end of the molding cycle and, thus, will form ahardened skin on the bottom surface of the molded article.

Example 2

Assume for purposes of this example that bottom mold 18 is formed fromsilicone rubber V-1075 (manufactured by Rhodia Inc.), which containsiron oxide and thus has a higher thermal conductivity than the siliconerubber V-1008 used in Example 1. Specifically, the thermal conductivityof silicone rubber V-1075 is 0.0241. All of the calculations and valuesof Example 1 remain unchanged, with the exception of the followinganalysis.

The temperature at the interface between foam material 20 and bottommold 18 may be calculated from equation (19) using the temperature (7′₃)of bottom mold 18 derived in Example 1 above and the value for thetemperature (T₂) of foam material 20 shown in Table 1 (assuming that thethermal conductivity of foam material 20 is still 0.00638 while thethermal conductivity of bottom mold 18 is now 0.0241):

$T_{2,3} = {\frac{\left( {{.00638} \times 200} \right) + \left( {{.0241} \times 55.48} \right)}{{.00638} + {.0241}} = {85.73{^\circ}\mspace{14mu}{C.}}}$

Thus, the temperature at the interface between foam material 20 andbottom mold 18 is now 85.73° C.

Equation (20) may then be used to determine the slope of temperatureline 24 using the temperature (T_(2,3)) at the interface between foammaterial 20 and bottom mold 18 derived above and the values for thetemperature (T₂) of foam material 20 and the thickness (d₂) of foammaterial 20 shown in Table 1:

${slope} = {\frac{200 - 85.73}{.1875} = {609.44{^\circ}\mspace{14mu}{{C.}/{inch}}}}$

Thus, the slope of temperature line 24 is now 609.44° C./inch.

Now, the thickness (x) of foam material 20 that remains under thedecomposition temperature (T_(B)) of the blowing agent may be calculatedfrom equation (21) using the temperature (T_(2,3)) at the interfacebetween foam material 20 and bottom mold 18 derived above and the slopeof temperature line 24 derived above (again, assuming that thedecomposition temperature (T_(B)) of the blowing agent in foam material20 is 150° C.):

$x = {\frac{150 - 85.73}{609.44} = {0.105\mspace{14mu}{inches}}}$

Thus, the thickness (x) of foam material 20 that remains under thedecomposition temperature (T_(B)) of the blowing agent is now 0.105inches. As such, a layer of foam material 20 having a thickness of 0.105inches will not be blown at the end of the molding cycle and, thus, willform a hardened skin on the bottom surface of the molded article.

In comparing this example to Example 1, it can be seen that increasingthe thermal conductivity (k₃) of bottom mold 18 functions to increasethe thickness of the hardened skin formed on the bottom surface of themolded article. Alternatively, as discussed above, the thickness of thehardened skin could also be increased by decreasing the power factor(pf₃) of bottom mold 18 or decreasing the temperature of bottomelectrode 14.

Example 3

In this example, equation (25) (which is a combination of equations (8)and (24)) will be used to calculate more exact values for the moldingtime (t₂) of foam material 20, the temperature (T₃) of bottom mold 18,and the thickness (x) of foam material 20 that remains under thedecomposition temperature (T_(B)) of the blowing agent (as compared tothe values calculated in Example 1 above). It should be understood thatthe subscripts used in the following equations denote the layer numberof the particular material (i.e., subscript 1 denotes top mold 16 (layer1), subscript 2 denotes foam material 20 (layer 2), subscript 3 denotesbottom mold 18 (layer 3), and subscript 4 denotes bottom electrode 14(layer 4)).

The power (P₂) that is dissipated in foam material 20 due to theapplication of the dielectric field may be calculated from equation (8)using the current (I) and capacitance (C₂) derived in Example 1 aboveand the value for the power factor (pf₂) shown in Table 1 (assuming thatthe frequency of the electric field is 40 MHz):

$P_{2} = {\frac{0.137 \times {.959}^{2}}{2 \times \pi \times 40 \times 10^{6} \times 1.599 \times 10^{12}} = {31.352\mspace{14mu}{watts}}}$

The power (Q₂) that is generated in foam material 20 due to the heatexchange from top mold 16 and bottom mold 18 may be calculated fromequation (24) using the thickness (d₂) of foam material 20 shown inTable 1:

$Q_{2} = {\frac{2}{.375}\left\lbrack {{\frac{k_{1} \times k_{2}}{k_{1} + k_{2}}\left( {T_{1} - T_{2}} \right)} + {\frac{k_{2} \times k_{3}}{k_{2} + k_{3}}\left( {T_{3} - T_{2}} \right)}} \right\rbrack}$

As discussed above, top mold 16 is heated at the same rate as foammaterial 20 such that T₁−T₂=0. Thus, the above equation may besimplified as follows (assuming that the thermal conductivity of foammaterial 20 is 0.00638 and the thermal conductivity of bottom mold 18 is0.00585):

$Q_{2} = {{5.333\left\lbrack {\frac{{.00585} \times {.00638}}{{.00585} + {.00638}}\left( {T_{3} - T_{2}} \right)} \right\rbrack} = {{.01627}{\left( {T_{3} - T_{2}} \right).}}}$

Similarly, the power (P₃) that is dissipated in bottom mold 18 due tothe application of the dielectric field may be calculated from equation(8) using the current (I) and capacitance (C₃) derived in Example 1above and the value for the power factor (pf₃) shown in Table 1 (again,assuming that the frequency of the electric field is 40 MHz):

$P_{3} = {\frac{{.0033} \times {.959}^{2}}{2 \times \pi \times 40 \times 10^{6} \times 4.72 \times 10^{- 12}} = {2.558\mspace{14mu}{watts}}}$

The power (Q₃) that is generated in bottom mold 18 due to the heatexchange from foam material 20 and bottom electrode 14 may be calculatedfrom equation (24) using the thickness (d₃) of bottom mold 18 shown inTable 1:

$Q_{3} = {\frac{2}{.125}\left\lbrack {{\frac{k_{2} \times k_{3}}{k_{2} + k_{3}}\left( {T_{2} - T_{3}} \right)} + {\frac{k_{3} \times k_{4}}{k_{3} + k_{4}}\left( {T_{4} - T_{3}} \right)}} \right\rbrack}$

As discussed above, bottom electrode 14 is chilled and maintained at 0°C. throughout the molding cycle such that T₄=0. Also, because thethermal conductivity (k₄) of bottom electrode 14 is very large comparedto the thermal conductivity (k₃) of bottom mold 18, the above equationmay be simplified as follows (again, assuming that the thermalconductivity of foam material 20 is 0.00638 and the thermal conductivityof bottom mold 18 is 0.00585):

$\begin{matrix}{Q_{3} = {\frac{2}{.125}\left\lbrack {{\frac{{.00638} \times {.00585}}{{.00638} + {.00585}}\left( {T_{2} - T_{3}} \right)} + {{.00585}\left( {0 - T_{3}} \right)}} \right\rbrack}} \\{= {{0.0488T_{2}} - {0.0936T_{3}}}}\end{matrix}$

Now, the time (Δt) required for foam material 20 to reach its moldingtemperature of 200° C. may be expressed by the following equation usingthe power (P₂) and (Q₂) derived above and the values for the thickness(d₂), specific heat (h₂), and specific gravity (ρ₂) shown in Table 1:

$\begin{matrix}\begin{matrix}{{\Delta\; t} = \frac{\Delta\; T_{2} \times h_{2} \times \rho_{2} \times d_{2} \times 16.387}{P_{2} + Q_{2}}} \\{= \frac{\Delta\; T_{2} \times 1.566 \times 1.0408 \times {.375} \times 16.387}{31.352 + {{.01627}\left( {T_{3} - T_{2}} \right)}}}\end{matrix} & (26)\end{matrix}$

Next, the increase in the temperature (ΔT₃) of bottom mold 18 may beexpressed by the following equation using the power (P₃) and (Q₃)derived above and the values for the thickness (d₃), specific heat (h₃),and specific gravity (ρ₃) shown in Table 1:

$\begin{matrix}\begin{matrix}{{\Delta\; T_{3}} = \frac{\Delta\;{t\left( {P_{3} + Q_{3}} \right)}}{h_{3} \times \rho_{3} \times d_{3} \times 16.387}} \\{= \frac{\Delta\;{t\left( {2.558 + {{.0488}T_{2}} - {{.0936}T_{3}}} \right)}}{1.509 \times 1.105 \times {.125} \times 16.387}} \\{= {\Delta\;{t\left( {{.7489} + {\ldots\mspace{14mu} 01429T_{2}} - {{.0274}T_{3}}} \right)}}}\end{matrix} & (27)\end{matrix}$

Equations (26) and (27) will now be calculated at 1 second timeintervals in order to calculate more exact values for the molding time(t₂) of foam material 20, the temperature (T₃) of bottom mold 18, andthe thickness (x) of foam material 20 that remains under thedecomposition temperature (T_(B)) of the blowing agent.

At the beginning of the molding cycle, the temperature (T₂) of foammaterial 20 is 0° C. and the temperature (T₃) of bottom mold 18 is 0° C.Thus, after a 1 second time interval, equation (26) may be expressed asfollows:

$1 = \frac{\Delta\; T_{2} \times 1.566 \times 1.0408 \times {.375} \times 16.387}{31.352 + {{.01627}\left( {0 - 0} \right)}}$This equation may then be solved such that ΔT₂=3.13° C. As such, thetemperature (T₂) of foam material 20 after 1 second of heating is 3.13°C. Equation (27) may also be expressed as follows:ΔT ₃=1(0.7489+0.01429(0)+0.0274(0))This equation may then be solved such that ΔT₃=0.7489° C. As such, thetemperature (T₃) of bottom mold 18 after 1 second of heating is 0.7489°C.

Now, after another 1 second time interval, equation (26) may beexpressed as follows:

$1 = \frac{\Delta\; T_{2} \times 1.566 \times 1.0408 \times {.375} \times 16.387}{31.352 + {{.01627}\left( {{.7489} - 3.13} \right)}}$This equation may then be solved such that ΔT₂=3.126° C. As such, thetemperature (T₂) of foam material 20 after 2 seconds of heating is6.250° C. Equation (27) may also be expressed as follows:ΔT ₃=1(0.7489+0.01429(3.13)+0.0274(0.7489))This equation may then be solved such that ΔT₃=0.7731° C. As such, thetemperature (T₃) of bottom mold 18 after 2 seconds of heating is 1.522°C.

It should be understood that the above steps may be repeated at every 1second time interval until the temperature (T₂) of foam material 20reaches 200° C. By doing so, it can be determined that the molding time(t₂) of foam material 20 is 66 seconds (compared to 63.54 seconds ascalculated in Example 1). Also, after 66 seconds of heating, thetemperature (T₃) of bottom mold 18 is 78.83° C. (compared to 55.48° C.as calculated in Example 1). In addition, the temperature (T_(2.3)) atthe interface of foam material 20 and bottom mold 18 after 66 seconds ofheating is 142° C. (compared to 130.87° C. as calculated in Example 1).As such, the thickness (x) of foam material 20 that remains under thedecomposition temperature (T_(B)) of the blowing agent is 0.0258 inches(compared to 0.0519 inches as calculated in Example 1).

Thus, it can be seen that calculating equations (26) and (27) at regulartime intervals results in more accurate values for the molding time (t₂)of foam material 20, the temperature (T₃) of bottom mold 18 at the endof the molding cycle, and the thickness (x) of foam material 20 thatremains under the decomposition temperature (T_(B)) of the blowingagent.

SECOND EXEMPLARY EMBODIMENT

Referring to FIG. 6, a second exemplary embodiment of the method of thepresent invention will now be described with reference to the diagram ofa flow molding apparatus designated generally as numeral 100, which isused to form the bottom sole of a tennis shoe or other athleticfootwear. It should be understood that all of the equations set forthabove in connection with the first exemplary embodiment also apply tothe second exemplary embodiment, in which two different thicknesses ofhardened skin are formed on the bottom surface of the shoe sole.

Flow molding apparatus 100 includes a top electrode 120 and a bottomelectrode 140, both of which are connected to an electromagnetic energysource (not shown) operable to generate an alternating electric fieldbetween the electrodes. The alternating electric field may be generatedat frequencies ranging from 1 MHz to 500 MHz, is preferably generated atfrequencies ranging from 10 MHz to 100 MHz, and is most preferablygenerated at either 26 MHz or 40 MHz.

Also included within apparatus 100 are a top mold 160 and a bottom mold180 that together define a molding cavity therebetween. As can be seenin FIG. 6, the top mold 160 is configured to define the shape of the topsurface of the shoe sole, and the bottom mold 180 is configured todefine the shape of the bottom surface of the shoe sole. The bottom mold180 is formed of various mold sections 180 a, 180 b, 180 c, 180 d, 180e, 180 f, 180 g, 180 h and 180 i that together form a unitary piece. Inthe illustrated example, a foam material 200 is placed within themolding cavity such that a top surface of foam material 200 is incontact with top mold 160 and a bottom surface of foam material 200 isin contact with the various mold sections of bottom mold 180. Inoperation, an alternating dielectric field is applied across foammaterial 200 to thereby form the molded article.

In accordance with the method of the present invention, a firstthickness of hardened skin is formed on the foam material 200 in contactwith mold section 180 a, and a second thickness of hardened skin isformed on the foam material 200 in contact with mold sections 180 b-180i. The first and second thicknesses of hardened skin are showncollectively as numeral 200 a in FIG. 6 and, as can be seen, are formedon a horizontal wall, on a slope, and on a vertical wall of the bottommold 180. In this embodiment, a hardened skin is not desired on the foammaterial 200 in contact with top mold 160 so as to provide cushioningalong the top surface of the shoe sole. Of course, it should beunderstood that one or more thicknesses of hardened skin could also beformed on the top surface of foam material 200 if desired (i.e., inother embodiments).

An example will now be provided with reference to the flow moldingapparatus of FIG. 6 to further describe the method of the presentinvention. It should be understood that this example is provided merelyto illustrate the manner in which the method may be used to manufacturea molded article with two or more thicknesses of hardened skin and doesnot in any way limit the scope of the present invention.

Example 4

Assume for purposes of this example that top mold 160 and bottom mold180 of flow molding apparatus 100 are each formed from liquid siliconerubber and foam material 200 comprises an EVA foam blend (wherein thedecomposition temperature of the blowing agent is approximately 310°F.). Table 2 is provided below to show various factors for each of theselayers of material, namely, the desired temperature (T) at the end ofthe molding cycle, power factor (pf), relative dielectric constant (∈),specific heat (h), and specific gravity (ρ). These factors will be usedhereinbelow to perform various calculations in accordance with themethod of the present invention. It should also be noted that thealternating electric field generated between the electrodes in thisexample is 40 MHz.

TABLE 2 Foam Top Mold Material Bottom Mold (layer 1) (layer 2) (layer 3)Desired Temperature 380 380 (T) (degrees Fahrenheit) Power Factor (pf).003 .028 .003 Relative Dielectric 3.07 3.07 3.07 Constant (ε) SpecificHeat (h) 1.233 1.34 1.233 Specific Gravity (ρ) 1.16 1.041 1.16

As can be seen from Table 2, the desired temperature of foam material200 at the end of the molding cycle is 380° F. (i.e., higher than thedecomposition temperature of the blowing agent so that it will be fullyblown and cross-linked). In this example, it is desired that the topsurface of the shoe sole be fully blown at the end of the molding cycle.As such, the desired temperature of the bottom surface of top mold 160at the end of the molding cycle is also 380° F. On the other hand, twodifferent thicknesses of hardened skin are desired on the bottom surfaceof the shoe sole at the end of the molding cycle. As such, bottomelectrode 140 is chilled throughout the molding cycle so that thetemperatures of the various mold sections of bottom mold 180 aresubstantially reduced. The exact temperatures of these various moldsections will be calculated hereinbelow.

Table 2 also shows the various values for the power factor (pf),relative dielectric constant (∈), specific heat (h) and specific gravity(ρ) for each of the layers of material. As discussed above in connectionwith the first exemplary embodiment, the values for the power factor(pf), relative dielectric constant (∈) and specific heat (h) vary withtemperature. It should be understood that the value-temperature curves(similar to those shown in FIGS. 3-5) were each integrated over theentire temperature range and the resultant averages are shown in Table2, which will be used below to provide a close approximation of thefollowing the molding time for foam material 200, the required powerfactor for top mold 160, the increase in temperature of bottom moldsection 180 a, and the required power factor for mold sections 180 b-180i. It should be understood that equations (25)-(27) may be used tocalculate more exact values as discussed above in connection withExample 3 of the first exemplary embodiment, wherein a computer may beprogrammed to perform these calculations in order to simplify theanalysis.

Notably, the relative dielectric constants (∈) of top mold 160, foammaterial 200 and bottom mold 180 are equalized (as shown in Table 2) soas to achieve uniform heating of foam material 200. Because thedielectric constants (s) have been equalized, the capacitance (C) willbe the same in every section of the mold (see equation (1) above) and,as a result, the equivalent reactance (X_(eq)) associated with theequivalent capacitance (C_(eq)) of all three layers of material will bethe same in every section of the mold (see equation (3) above). Assumingthat the equivalent resistance (R_(eq)) of all three layers of materialis small compared to the equivalent reactance (X_(eq)), it follows thatthe current (I) will be the same in every section of the mold (seeequation (6) above).

Now, by combining equations (1) and (12) above, the power factor foreach layer of material may be expressed as follows:

$\begin{matrix}{{pf}_{i} = {\frac{16.387 \times \Delta\; T_{i} \times h_{i} \times \rho_{i} \times d_{i} \times 2 \times \pi \times f}{t_{i} \times I^{2}} \times \frac{25.4 \times ɛ_{i} \times A_{i}}{36 \times \pi \times d_{i}}}} & (26)\end{matrix}$

If we study the power factor for a unit of area of 1 inch², equation(26) may be simplified as follows:

$\begin{matrix}{{pf}_{i} = {\frac{\Delta\; T_{i} \times h_{i} \times \rho_{i}}{t_{i}} \times \frac{16.387 \times 25.4 \times ɛ_{i} \times 2 \times \pi \times f}{36 \times \pi \times I^{2}}}} & (27)\end{matrix}$

As discussed above, the dielectric constant (∈) and the current (I) arethe same in every section of the mold (wherein/may be calculated usingequations (1)-(5) above). As such, equation (27) may be expressed asfollows:

$\begin{matrix}{{pf}_{i} = {\frac{\Delta\; T_{i} \times h_{i} \times \rho_{i}}{t_{i}} \times {constant}}} & (28)\end{matrix}$wherein, in this example, the constant is 1.902×10⁻³.

Equation (28) may be solved for t_(i) such that the molding time oflayer i may be expressed as follows:

$\begin{matrix}{t_{i} = \frac{\Delta\; T_{i} \times h_{i} \times \rho_{i} \times 1.902 \times 10^{- 3}}{{pf}_{i}}} & (29)\end{matrix}$

Equation (28) may also be solved for ΔT_(i) such that the increase intemperature of layer i may be expressed as follows:

$\begin{matrix}{{\Delta\; T_{i}} = \frac{t_{i} \times {pf}_{i}}{h_{i} \times \rho_{i} \times 1.902 \times 10^{- 3}}} & (30)\end{matrix}$

In accordance with the method of the present invention, equations(28)-(30) may be used to calculate: (1) the molding time of foammaterial 200; (2) the required power factor for top mold 160 so that thetop surface of foam material 200 will be fully blown and preferablycross-linked at the end of the molding cycle; (3) the increase intemperature of bottom mold section 180 a so as to form a thick skin oncertain bottom surfaces of foam material 200 at the end of the moldingcycle; and (4) the required power factor for mold sections 180 b-180 iso as to form a thin skin on certain bottom surfaces of foam material200 (i.e., the heel and recessed portions) at the end of the moldingcycle. In these equations, subscript 1 denotes top mold 160, subscript 2denotes foam material 200, subscript 3 denotes bottom mold section 180a, and subscript 4 denotes bottom mold sections 180 b-180 i.

First, equation (29) may be used to calculate the molding time of foammaterial 200 using the values for the specific heat (h), specificgravity (ρ) and power factor (pf) shown in Table 2 (assuming that thestarting temperature is 70° F. ambient temperature):

$\begin{matrix}{t_{2} = \frac{\Delta\; T_{2} \times h_{2} \times \rho_{2} \times 1.902 \times 10^{- 3}}{{pf}_{2}}} \\{= \frac{310 \times 1.34 \times 1.041 \times 1.902 \times 10^{- 3}}{.028}} \\{= {29.4\mspace{14mu}{seconds}}}\end{matrix}$Thus, the molding time for foam material 200 is 29.4 seconds (i.e., thetime that it takes foam material 200 to reach 380° F.).

Second, the molding time (t₂) for foam material 200 (as calculatedabove) is used to calculate a required power factor (pf₁) for top mold160. As stated above, the temperature of top mold 160 is desirablychosen so that the top surface of foam material 200 will be fully blownat the end of the molding cycle. As such, the required power factor(pf₁) for top mold 160 will be the power factor that allows the bottomsurface of top mold 160 to reach a temperature that is substantially thesame as the molding temperature of foam material 200 at the end of themolding cycle. The required power factor (pf₁) for top mold 160 may becalculated from equation (28) using the molding time (t) for foammaterial 200 and the values for the specific heat (h) and specificgravity (ρ) shown in Table 2 (again, assuming that the startingtemperature is 70° F. ambient temperature):

$\begin{matrix}{{pf}_{1} = {\frac{\Delta\; T_{1} \times h_{1} \times \rho_{1}}{t_{2}} \times 1.902 \times 10^{- 3}}} \\{= {\frac{310 \times 1.233 \times 1.16}{29.4} \times 1.902 \times 10^{- 3}}} \\{= 0.0286}\end{matrix}$

Thus, the required power factor for top mold 160 is 0.0286. It should beunderstood that the power factor of top mold 160 may be adjusted byselecting a polar additive, calculating the amount of the polar additiveto be mixed with the material of top mold 160 so that the power factormatches the required power factor derived above (using equation (15)above), and then mixing the calculated amount of the polar additive withthe material of top mold 160.

Third, the molding time (t₂) for foam material 200 (as calculated above)is used to calculate the increase in temperature (ΔT₃) of bottom moldsection 180 a at the end of the molding cycle. Preferably, bottomelectrode 140 is chilled throughout the molding cycle so that thetemperature of bottom mold section 180 a may be substantially reduced toenable the formation of a thick skin on the bottom surface of the shoesole as needed to withstand the abrasion that occurs during normal useof the shoe sole. Preferably, the portions with this thick skin arechosen to have as small of an area as possible in order to minimize theweight of the shoe sole. The increase in temperature (ΔT₃) of bottommold section 180 a may be calculated from equation (30) using thecalculated molding time (t₂) for foam material 200 and the values forthe power factor (pf), specific heat (h) and specific gravity (ρ) shownin Table 2:

$\begin{matrix}{{\Delta\; T_{3}} = \frac{t_{2} \times {pf}_{3}}{h_{3} \times \rho_{3} \times 1.902 \times 10^{- 3}}} \\{= \frac{29.4 \times {.003}}{1.233 \times 1.16 \times 1.902 \times 10^{- 3}}} \\{= {32.5\;{^\circ}\mspace{14mu}{F.}}}\end{matrix}$

Thus, the temperature of bottom mold section 180 a at the end of themolding cycle is 102.5° F. (32.5° F.+70° F. ambient temperature), whichis quite cold and well below the decomposition temperature of theblowing agent. As such, mold section 180 a will cool by thermalconductivity the portion of foam material 200 in contact therewith so asto form a thick skin on the bottom surface of the shoe sole. It shouldbe understood that the thickness of the skin may be calculated usingequation (21) above.

Fourth, the molding time (t₂) for foam material 200 (as calculatedabove) is used to calculate a required power factor (pf₄) for bottommold sections 180 b-180 i. The temperature of bottom mold sections 180b-180 i is desirably chosen to be 260° F., which is 50° F. below thedecomposition temperature of the blowing agent and will cool by thermalconductivity the portion of foam material 200 in contact therewith so asto form a thin skin as needed to prevent tears and provide anaesthetically pleasing “look” for the shoe sole. The required powerfactor (pf₄) for bottom mold sections 180 b-180 i may be calculated fromequation (28) using the molding time (t) for foam material 200 and thevalues for the specific heat (h) and specific gravity (ρ) shown in Table2 (again, assuming that the starting temperature is 70° F. ambienttemperature):

$\begin{matrix}{{pf}_{4} = {\frac{\Delta\; T_{4} \times h_{4} \times \rho_{4}}{t_{2}} \times 1.902 \times 10^{- 3}}} \\{= {\frac{190 \times 1.233 \times 1.16}{29.4} \times 1.902 \times 10^{- 3}}} \\{= 0.0177}\end{matrix}$

Thus, the required power factor for bottom mold sections 180 b-180 i is0.0177. It should be understood that bottom mold sections 180 b-180 imay be adjusted by selecting a polar additive, calculating the amount ofthe polar additive to be mixed with the material of these bottom moldsections so that the power factor matches the required power factorderived above (using equation (15) above), and then mixing thecalculated amount of the polar additive with the material of thesebottom mold sections. It should also be understood that the thickness ofthe skin formed on bottom mold sections 180 b-180 i may be calculatedusing equation (21) above.

It should be understood that flow molding apparatus 100 is merely anexample of an apparatus that may be used to make a molded article withdifferent thicknesses of hardened skin in accordance with the method ofthe present invention. Other flow molding apparatuses and relatedmethods may also be used, whereby a hardened skin of any desiredthickness may be obtained on one or more surfaces (sides) of the moldedarticle or on only a portion of a particular surface. The thickness ofthe hardened skin may vary between the various surfaces, and may becontrolled by varying the temperature of the appropriate mold sections.In general, a high temperature for a particular mold section will resultin no skin, a medium temperature for a particular mold section willresult in a thin skin, and a low temperature for a particular moldsection will result in a thick skin (wherein the high, medium and lowtemperatures are determined in relation to the decomposition temperatureof the blowing agent).

Preferably, the temperature of each of the various mold sections (andthus the thickness of the skin) is controlled by mixing an appropriateamount of additive with the material of the mold section (e.g., anadditive mixed with liquid silicone rubber), unless a thick skin isdesired in which case no additive is used. Alternatively, the thicknessof the skin may be controlled by one or more additional means,including:

-   -   (a) changing the thickness of a mold section (in the case where        a mold section is heated, a thin mold section will result in a        thicker skin than a thick mold section; in the case where a mold        section remains cold, a thin mold section will result in a        thicker skin than a thick mold section);    -   (b) changing the thermal conductivity of a mold section (mold        compounds with a higher thermal conductivity will result in a        thicker skin than mold compounds with a lower thermal        conductivity);    -   (c) chilling one of the electrodes to a lower temperature than        the other electrode (a chilled electrode with a lower the        temperature will result in a thicker skin than a chilled        electrode with a higher temperature);    -   (d) using a three-dimensional electrode (a chilled electrode        section that is closer to the foam material will result in a        thicker skin than another chilled electrode section that is        further from the foam material);    -   (e) changing the dielectric constant of a mold section (mold        compounds with a higher dielectric constant will result in less        current and a thicker skin than mold compounds with a lower        dielectric constant); and    -   (f) any combination of the above.

As demonstrated above in connection with the first and second exemplaryembodiments, the method of the present invention may be used to make amolded article from a single foam material placed in a molding cavity ofa flow molding apparatus. In particular, a hardened skin may be formedon one or more surfaces (e.g., one or more portions of a top surface,bottom surface, side surface, etc.) of the molded article by reducingthe temperature of one or more sections of the molds that are in contactwith the foam material. Alternatively, the method may be used to make amolded article from two or more different formable materials (at leastone of which is a foam material) placed in a molding cavity of a flowmolding apparatus. Again, a hardened skin may be formed on the moldedarticle by reducing the temperature of one or more of the mold sectionsthat are in contact with the foam material.

In view of the foregoing, it should be understood that the method of thepresent invention enables the fabrication of molded articles withhardened skins that are sufficiently durable to withstand the abrasionthat occurs during normal use of the articles. The method may also beused to make molded articles with hardened skins that are easilywashable. The method may further be used to fabricate molded articleswith hardened skins that may be texturized as desired. In addition, themethod may be used to make molded articles with hardened skins that havehigher coefficients of friction to thereby provide non-skid surfaces asrequired for particular applications. Of course, other advantages of thepresent invention should be apparent to one skilled in the art.

While the present invention has been described and illustratedhereinabove with reference to various exemplary methods, it should beunderstood that various modifications could be made to this methodwithout departing from the scope of the invention. Therefore, theinvention is not to be limited to the exemplary methods described andillustrated hereinabove, except insofar as such limitations are includedin the following claims.

1. A method of making a molded article during a single heating cycle,comprising: providing a flow molding apparatus comprising a first moldsection and a second mold section; selecting a foam material from whichto form the molded article, wherein the foam material comprises aformable material mixed with a blowing agent; placing the foam materialin the flow molding apparatus such that a first surface of the foammaterial is in contact with the first mold section and a second surfaceof the foam material is in contact with the second mold section;adjusting a characteristic of the second mold section so as to control athickness of hardened skin to be formed on the second surface of thefoam material; and heating the foam material in the flow moldingapparatus by applying an alternating electric field across the foammaterial such that, at the end of the heating cycle, (i) the firstsurface of the foam material exceeds a decomposition temperature of theblowing agent and is blown and (ii) the second surface of the foammaterial remains under the decomposition temperature of the blowingagent and is not blown to thereby form the thickness of hardened skin onthe second surface of the foam material.
 2. The method of claim 1,wherein the thickness of the hardened skin is controlled by changing thethickness of the second mold section, changing the thermal conductivityof the second mold section, adjusting the temperature of an electrode ofthe flow molding apparatus, utilizing a three-dimensional electrode,changing the dielectric constant of the second mold section, or anycombination of the foregoing.
 3. The method of claim 1, wherein the flowmolding apparatus comprises a third mold section, wherein the foammaterial is placed in the flow molding apparatus such that a thirdsurface of the foam material is in contact with the third mold section,and wherein the third surface of the foam material remains under thedecomposition temperature of the blowing agent and is not blown at theend of the single heating cycle so as to form a thickness of hardenedskin on the third surface of the foam material.
 4. The method of claim3, wherein the thickness of hardened skin on the second surface of thefoam material is not equal to the thickness of hardened skin on thethird surface of the foam material.
 5. The method of claim 1, whereinthe first and second mold sections comprise a top mold and a bottommold, respectively, of the flow molding apparatus.
 6. The method ofclaim 1, wherein the first and second mold sections together comprise atop mold or a bottom mold of the flow molding apparatus.
 7. A method ofmaking a molded article during a single heating cycle, comprising:providing a flow molding apparatus comprising a first mold section and asecond mold section; selecting a foam material from which to form themolded article, wherein the foam material comprises a formable materialmixed with a blowing agent; calculating a molding time for the foammaterial; calculating a required power factor for the first mold sectionbased on the calculated molding time of the foam material; adjusting thepower factor of the first mold section so as to substantially match therequired power factor; placing the foam material in the flow moldingapparatus such that a first surface of the foam material is in contactwith the first mold section and a second surface of the foam material isin contact with the second mold section; and heating the foam materialto form the molded article such that, at the end of the heating cycle,(i) the first surface of the foam material exceeds a decompositiontemperature of the blowing agent and is blown and (ii) the secondsurface of the foam material remains under the decomposition temperatureof the blowing agent and is not blown so as to form a thickness ofhardened skin on the second surface of the foam material.
 8. The methodof claim 7, wherein the foam material is heated by applying analternating electric field across the foam material.
 9. The method ofclaim 7, wherein the thickness of hardened skin is controlled bychanging the thickness of the second mold section, changing the thermalconductivity of the second mold section, adjusting the temperature of anelectrode of the flow molding apparatus, utilizing a three-dimensionalelectrode, changing the dielectric constant of the second mold section,or any combination of the foregoing.
 10. The method of claim 7, whereinthe flow molding apparatus comprises a third mold section, wherein thefoam material is placed in the flow molding apparatus such that a thirdsurface of the foam material is in contact with the third mold section,and wherein the third surface of the foam material remains under thedecomposition temperature of the blowing agent and is not blown at theend of the single heating cycle so as to form a thickness of hardenedskin on the third surface of the foam material.
 11. The method of claim10, wherein the thickness of hardened skin on the second surface of thefoam material is not equal to the thickness of hardened skin on thethird surface of the foam material.
 12. The method of claim 7, whereinthe first and second mold sections comprise a top mold and a bottommold, respectively, of the flow molding apparatus.
 13. The method ofclaim 7, wherein the first and second mold sections together comprise atop mold or a bottom mold of the flow molding apparatus.